Peacocks nearby: approximating sequences of measures

A peacock is a family of probability measures with finite mean that increases in convex order. It is a classical result, in the discrete time case due to Strassen, that any peacock is the family of one-dimensional marginals of a martingale. We study the problem whether a given sequence of probability measures can be approximated by a peacock. In our main results, the approximation quality is measured by the infinity Wasserstein distance. Existence of a peacock within a prescribed distance is reduced to a countable collection of rather explicit conditions. This result has a financial application (developed in a separate paper), as it allows to check European call option quotes for consistency. The distance bound on the peacock than takes the role of a bound on the bid-ask spread of the underlying. We also solve the approximation problem for the stop-loss distance, the L\'evy distance, and the Prokhorov distance.